Projective curve/Canonical extension/Properties/Section

Let ${}C$ be a smooth projective curce over an algebraically closed field ${}K$ of genus ${}g$. We look at the short exact sequence

$0\,{\stackrel {}{\longrightarrow }}\,{\mathcal {O}}_{C}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {E}}=\operatorname {Der} _{R}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {B}}=\operatorname {Der} _{C}\,{\stackrel {}{\longrightarrow }}\,0.$

For ${}g\geq 1$, we have

${}\operatorname {deg} {\left(\operatorname {Der} _{C}\right)}\leq 0\,,$

and the locally free sheaf in the middle is not stable; for ${}g\geq 2$, it is not even semistable. We work with

$0\,{\stackrel {}{\longrightarrow }}\,\operatorname {Sym} ^{q-1}{\left({\mathcal {E}}\right)}\,{\stackrel {}{\longrightarrow }}\,\operatorname {Sym} ^{q}{\left({\mathcal {E}}\right)}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {B}}^{q}\,{\stackrel {}{\longrightarrow }}\,0,$

and tensorizations whereof. In characteristic zero, this sequence does not split, because then the symmetric powers can be embedded into the tensor powers.

Lemma

Let ${}C$ denote a smooth projective curve over an algebraically closed field ${}K$ of characteristic ${}0$, and let

$0\,{\stackrel {}{\longrightarrow }}\,{\mathcal {O}}_{C}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {E}}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {B}}\,{\stackrel {}{\longrightarrow }}\,0$

be a non-trivial extension, where ${}{\mathcal {B}}$ is invertible. Let ${}{\mathcal {H}}$ denote a very ample invertible sheaf. Set

${}r=-{\frac {\operatorname {deg} {\left({\mathcal {B}}\right)}}{\operatorname {deg} {\left({\mathcal {H}}\right)}}}\,.$

Then for

${}m\leq rq$,

the global sections of ${}\operatorname {Sym} ^{q}{\left({\mathcal {E}}\right)}\otimes {\mathcal {H}}^{m}$ come from ${}\operatorname {Sym} ^{q-1}{\left({\mathcal {E}}\right)}\otimes {\mathcal {H}}^{m}$.

Proof

We look at

$0\,{\stackrel {}{\longrightarrow }}\,\operatorname {Sym} ^{q-1}{\left({\mathcal {E}}\right)}\otimes {\mathcal {H}}^{m}\,{\stackrel {}{\longrightarrow }}\,\operatorname {Sym} ^{q}{\left({\mathcal {E}}\right)}\otimes {\mathcal {H}}^{m}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {B}}^{q}\otimes {\mathcal {H}}^{m}\,{\stackrel {}{\longrightarrow }}\,0.$

Suppose that ${}m$ fulfills the numerical condition. Then the degree of ${}{\mathcal {B}}^{q}\otimes {\mathcal {H}}^{m}$ is ${}\leq 0$. If this is negative, then this sheaf does not have global sections, and the statement is clear. If the degree is ${}0$ and this sheaf is not the structure sheaf, then the statement is also clear. So suppose it is the structure sheaf. Since this sequence does not split, the nontrivial section of the structure sheaf is mapped to the non-zero cohomology class describing the extension. But then again, the first mapping is a bijection.

$\Box$

Lemma

Let ${}C$ denote a smooth projective curve of genus ${}g\geq 1$ over an algebraically closed field ${}K$, and let

$0\,{\stackrel {}{\longrightarrow }}\,{\mathcal {O}}_{C}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {E}}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {B}}\,{\stackrel {}{\longrightarrow }}\,0$

be an extension, where ${}{\mathcal {B}}$ is invertible of degree ${}\leq 0$. Let ${}{\mathcal {H}}$ denote a very ample invertible sheaf. Set

${}r=-{\frac {\operatorname {deg} {\left({\mathcal {B}}\right)}}{\operatorname {deg} {\left({\mathcal {H}}\right)}}}\,$

and let ${}c>\operatorname {deg} {\left(\omega _{C}\right)}$ be such that every invertible sheaf with degree above ${}c$ has nontrivial global sections. Then for ${}m\geq rq+c$,

there exist global sections of ${}\operatorname {Sym} ^{q}{\left({\mathcal {E}}\right)}\otimes {\mathcal {H}}^{m}$ that do not come from ${}\operatorname {Sym} ^{q-1}{\left({\mathcal {E}}\right)}\otimes {\mathcal {H}}^{m}$.

Proof

We look at

$0\,{\stackrel {}{\longrightarrow }}\,\operatorname {Sym} ^{q-1}{\left({\mathcal {E}}\right)}\otimes {\mathcal {H}}^{m}\,{\stackrel {}{\longrightarrow }}\,\operatorname {Sym} ^{q}{\left({\mathcal {E}}\right)}\otimes {\mathcal {H}}^{m}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {B}}^{q}\otimes {\mathcal {H}}^{m}\,{\stackrel {}{\longrightarrow }}\,0$

under the given numerical condition. Then the first cohomology group of the sheaf on the left is ${}0$ and ${}{\mathcal {B}}^{q}\otimes {\mathcal {H}}^{m}$ has nontrivial global sections, which come from the sheaf in the middle. These sections can not come from the left.

$\Box$

Theorem

Let ${}R$ denote a ${}2$-dimensional normal standard-graded over an algebraically closed field ${}K$ of characteristic ${}0$ and suppose that the genus of the corresponding smooth projective curve is at least ${}1$. Then the ${}R$-algebra

${}\bigoplus _{q}{{\left(\operatorname {Sym} ^{q}{\left(\operatorname {Der} \right)}\right)}^{*}}^{*}=\bigoplus _{q}{\left(\operatorname {Sym} ^{q}{\left(\Omega _{R{|}K}\right)}\right)}^{*}\,$

is not finitely generated. This means that the ring of global sections of the cotangent bundle above the smooth locus of ${}R$ is not finitely generated.

Proof

Assume that is generated by finitely many homomogenous elements of bidegree ${}(m_{j},q_{j})$ (with the exception of the Euler derivation ${}E$, which has degree ${}(0,1)$) satisfying ${}m_{j}>rq_{j}$ (generators that do not satisfy this condition come from the left, that is, they involve an ${}E$). Set

${}{\tilde {r}}=\operatorname {min} {\left({\frac {m_{j}}{q_{j}}}\right)}\,.$

We have then ${}{\tilde {r}}>r$. We consider in the ${}q-m$-plane the lines ${}rq$, ${}rq+c$ and ${}{\tilde {r}}q$. In the region above ${}rq+c$ and below ${}{\tilde {r}}q$, there are sections that are not generated by the finitely many given generators.

$\Box$